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Critical closed bosonic strings have independent left- and right-moving Virasoro constraints. Positive modes in both sectors annihilate physical kets, while both zero-mode conditions impose the mass shell and closed-string level matching. Quotient by physical null string states, equivalently use the appropriate BRST cohomology. With $L_0={\alpha }^{\prime }{k}^2/4+N$, the result is $N=\tilde{N}$ and $M^2=4(N-1)/{\alpha }^{\prime }$. Negative matter modes are adjoints rather than additional ket-annihilation constraints.

For a compact string coordinate of radius $R$, choose $p_L=n/R+wR/{\alpha }^{\prime }$ and $p_R=n/R-wR/{\alpha }^{\prime }$, where the left-moving oscillators are denoted by tildes. The two classical zero-mode Virasoro constraints give $M^2=p_L^2+4\tilde{N}/{\alpha }^{\prime }=p_R^2+4N/{\alpha }^{\prime }$. Their difference yields the displayed shifted closed-string level matching condition. Their average is $M^2=(n/R{)}^2+(wR/{\alpha }^{\prime }{)}^2+2(N+\tilde{N})/{\alpha }^{\prime }$. Classically the center-of-mass momentum can be continuous; the integer momentum number belongs to compact-coordinate quantization. Quantum intercepts shift the mass formula but cancel out of level matching.

The left and right transverse oscillator sectors each have generating function ${\prod }_{m\ge 1}(1-q^m{)}^{-24}$. Its coefficient satisfies $d_N\sim 2^{-1/2}(N-1{)}^{-27/4}{e}^{4\pi \sqrt{N-1}}$. Closed-string level matching requires equal levels but leaves their polarizations independent, so $g_N=d_N^2$. With ${\alpha }^{\prime }{M}^2=4(N-1)$, the exponential is $e^{{\beta }_{H}M}$ with ${\beta }_H=4\pi \sqrt{{\alpha }^{\prime }}$. The smooth rest-mass density is proportional to $M^{-26}{e}^{{\beta }_{H}M}$, since $dN/dM={\alpha }^{\prime }M/2$. This distinguishes degeneracy per level from density per unit mass. The polynomial prefactor matters at the boundary of canonical convergence.

The self-dual circle mass equation and closed-string level matching exhaust the possibilities at level sum zero, one or two. Besides the usual neutral $(N,\tilde{N})=(1,1)$ states, charges $(n,w)=(\pm 1,\pm 1)$ supply singly excited chiral states, with the sector fixed by level matching. Ground-state charges $(\pm 2,0)$ and $(0,\pm 2)$ also become massless. Their compact energy cancels the uncompactified tachyon mass. With $d$ physical transverse components, these families have $d^2+4d+4$ independent polarizations.

The Nambu–Goto action is invariant under worldsheet diffeomorphisms. Its canonical Nambu–Goto phase-space constraints areIn conformal gauge, $\Pi =T\dot{X}$, so these become $(\dot{X}\pm X^{\prime }{)}^2=0$. Their smeared generators implement the residual conformal transformations rather than independent propagating degrees of freedom. Classically their Fourier modes form the classical Virasoro constraint algebra. Quantization promotes them to the Virasoro algebra, and its constraints select the physical string states from the indefinite covariant Fock space.

For clarity, start with a Neumann boundary condition at both ends of an open string, $0\le \sigma \le \pi$. Its open-string mode expansion isCanonical quantization gives the string oscillator relationsWith ${\partial }_{\pm }=({\partial }_{\tau }\pm {\partial }_{\sigma })/2$, the worldsheet stress-energy tensor obeysSubstituting the open-string mode expansion into this quadratic expression derives the Virasoro generators:The colons mean normal ordering, placing the negative-mode creation operators to the left. The string level operator $N$ increases by $r$ when an oscillator ${\alpha }_{-r}$ is added.

Using the commutator once givesApplying this to the quadratic expression for $L_n$ produces $(m-n)L_{m+n}$, together with the double-contraction Virasoro central extension. To determine its coefficient, take $m>0$ and evaluate $[L_m,L_{-m}]$ on a zero-momentum oscillator vacuum. Only the creation pairs with $1\le r\le m-1$ contribute, givingHere $D$ is the target spacetime dimension; the timelike free boson also contributes one to the central charge, since a double contraction contains two metric factors. ThusThis derives both the Virasoro generators and their quantum anomaly.

In old covariant string quantization, the physical-state Virasoro conditions for an open string arewhere $a$ is the normal-ordering constant of a string. Requiring all positive and negative modes to annihilate a state would contradict the nonzero Virasoro central extension; the positive-mode prescription is the appropriate covariant subsidiary condition. The zero-mode equation gives the open bosonic string mass spectrum,For a closed string, there are two commuting Virasoro algebras. With uncompactified zero modes, $L_0={\alpha }^{\prime }{p}^2/4+N_R$ and ${\tilde{L}}_0={\alpha }^{\prime }{p}^2/4+N_L$. Imposing both positive-mode conditions and both zero-mode conditions gives the closed-string level matching $N_L=N_R$ and $M^2=4(N_L-a)/{\alpha }^{\prime }$.

Covariance initially retains the timelike string oscillators. For example, ${\alpha }_{-1}^0|0;p⟩$ has negative norm because ${\eta }^{00}=-1$. Its existence in the unconstrained space is not an inconsistency if it is absent from the physical quotient. The no-ghost theorem for the critical bosonic string states that, at $D=26$ and $a=1$, the inner product on physical string states at ordinary nonzero momentum is positive semidefinite. After quotienting physical null string states, this physical space is represented by the 24 positive-norm transverse string oscillators.

A spurious string state has the form ${\sum }_{n>0}{L}_{-n}|{\chi }_n⟩$. Its inner product with every physical string state vanishes, by $L_n|\psi ⟩=0$ and $L_n^{†}=L_{-n}$. Whenever such a state is itself physical, it is a null string state and represents a gauge redundancy, not an extra polarization. Consequently the physical quotient has no negative-norm propagating states, and agrees with light-cone gauge in string theory. This is the significance of the theorem; a proof is not needed here. It does not eliminate the bosonic tachyon, whose negative mass squared is distinct from negative norm, nor does it remove the auxiliary worldsheet ghost fields used in gauge fixing.

The Polyakov action introduces an independent worldsheet metric $h_{ab}$:Its redundancies are worldsheet diffeomorphisms and Weyl transformations. A Polyakov path integral over all metrics without dividing by this gauge volume would count physically equivalent configurations repeatedly. Fixing conformal gauge therefore needs the Faddeev-Popov determinant, not merely the substitution of a flat metric into the action.

After using the Weyl transformation to remove the trace of a metric variation, the infinitesimal worldsheet diffeomorphism ghost operator isThe Faddeev-Popov determinant of this operator is represented by anticommuting fields: a vector Faddeev-Popov ghost $c^a$ and a symmetric traceless Faddeev-Popov antighost field $b^{ab}$. In Euclidean coordinates their worldsheet ghost action is, up to a consistent overall normalization,The two chiral copies of this bc system have conformal weights $(2,-1)$ and anticommutators $\{b_m,c_n\}={\delta }_{m+n,0}$. They are auxiliary fields for the gauge determinant, not additional spacetime particles. Their use is different from the unwanted timelike negative-norm string states of covariant matter quantization.

The chiral holomorphic stress-energy tensor of the bc system isDouble contractions in its operator product expansion give the central charge of reparameterization ghosts $c_{\mathrm{gh}}=-26$. Equivalently, a fermionic bc system with antighost weight $\lambda$ has $c=1-3(2\lambda -1{)}^2$, and $\lambda =2$ gives $-26$. The free coordinate matter contributes $c_{\mathrm{m}}=D$. Thus the total worldsheet Weyl anomaly vanishes for the flat critical bosonic string theory precisely whenThis statement assumes no additional matter or compensating Liouville field theory.

BRST symmetry expresses the gauge symmetry after gauge fixing by replacing its infinitesimal parameter with the Grassmann variable $c$. In one chiral conformal patch, its graded action has the formthere is an analogous barred copy for a closed string. The Grassmann parity makes $s^{2}X=0$: the term from $sc$ cancels the term from applying $s$ to $\partial X$, while the term with $c^2$ vanishes. The gauge-fixed chiral transformations are on shell when their conservation equations are used. At the quantum level, BRST nilpotence also tests the anomaly.

For example, an open string BRST charge can be written using the matter Virasoro generators asUse the ghost oscillator vacuum $b_{n\ge 0}|0{⟩}_{mathrmgh}=c_{n>0}|0{⟩}_{mathrmgh}=0$ and the corresponding oscillator normal ordering. In this convention the last term is $-c_0$ in the critical theory. With a differently shifted ghost $L_0$, the displayed intercept must be shifted consistently; one must not count the same shift twice. Squaring $Q_B$ shows why both critical conditions are needed. Its anomalous terms, in this convention, have coefficientsThe independent cubic and linear terms vanish at $D=26$, $a=1$. The equivalent chiral BRST current is $j_B=c(T_{mathrmm}+T_{mathrmgh}/2)+(3/2){\partial }^{2}c$, with its contour integral defining the charge.

The graded identity $\{Q_B,b_n\}=L_n^{\mathrm{tot}}$ recovers the gauge constraints. In particular, on $|\psi ⟩\otimes |0{⟩}_{mathrmgh}$, BRST-closed imposes $L_{n>0}^{\mathrm{m}}|\psi ⟩=0$ and $(L_0^{\mathrm{m}}-1)|\psi ⟩=0$. Physical states are classes in BRST cohomology,at the appropriate ghost number: one for the usual open-string vertex and two for the closed-string vertex, with the closed-string zero-mode and closed-string level matching conditions also imposed. Since $Q_B^2=0$, an exact shift preserves closure. BRST Ward identities make exact insertions decouple from physical amplitudes and establish gauge independence. Together with the no-ghost theorem for the critical bosonic string, this explains how the auxiliary ghosts enforce gauge invariance while the physical spectrum retains positive norm.

Let $Y$ be the compact coordinate, identified modulo $2\pi R$. The closed string may wrap the circle:Single-valued momentum eigenfunctions $e^{ipY}$ require $p=n/R$, $n\in \mathbb{Z}$. These are the momentum and winding modes. Write the chiral zero-mode momenta asThe left-moving string oscillators will be tilded: $N_L=\tilde{N}$, $N_R=N$. This fixes the sign in compact-circle closed-string level matching.

Define the lower-dimensional mass by $M^2=-p_{\mathrm{noncompact}}^2$. The two closed-string physical-state Virasoro conditions, with bosonic string intercept one, areTheir average and difference derive the complete free bosonic string mass spectrum on the circle:The allowed states have nonnegative integer oscillator levels and obey the Virasoro constraints within each chiral sector. Compactification has not removed oscillator polarizations; it has added quantized compact momentum and allowed winding while changing the lower-dimensional interpretation of those polarizations.

At large $R$, the momentum and winding modes with $w=0$ have closely spaced momentum energies $|n|/R$, whereas nonzero winding costs $|w|R/{\alpha }^{\prime }$. At small $R$ their roles reverse. The precise equality is T-duality:It leaves $p_L$ unchanged and reverses $p_R$, and leaves the mass formula and closed-string level matching unchanged. On the string embedding map it reverses the right-moving part of $Y$. Therefore the small-radius spectrum is the large-radius spectrum of the dual theory, rather than a spectrum in which all charged states simply become heavy.

At a generic radius, the neutral level $(N_L,N_R)=(1,1)$ contains the lower-dimensional graviton, Kalb–Ramond field and dilaton, two gauge bosons from the mixed components of the metric and two-form, and the radius modulus of a string compactification. The two vector charges are the Cartan charges of $U(1{)}_L\times U(1{)}_R$.

Consider instead the following charged states:Every state satisfies compact-circle closed-string level matching. The single noncompact string oscillator supplies a vector polarization. At the massless point, the level-one Virasoro constraint imposes transverse polarization and its physical null string state identifies polarizations differing by the noncompact momentum. Thus these are genuine lower-dimensional vector states. Their squared mass isThus four additional charged massless spin-one particles appear atFor $(N_L,N_R)=(0,1)$ the charges have $p_R=0$, $p_L=\pm 2/\sqrt{{\alpha }^{\prime }}$; for $(1,0)$ they have $p_L=0$, $p_R=\pm 2/\sqrt{{\alpha }^{\prime }}$. This identifies the two pairs of charged roots.

The enhanced gauge group can be derived rather than just inferred from counting. At the self-dual circle, the chiral compact boson satisfies $Y_L(z)Y_L(u)\sim -({\alpha }^{\prime }/2)\log (z-u)$. The self-dual circle current algebra has currentsThe exponentials have conformal weight ${\alpha }^{\prime }{p}_L^2/4=1$. Free-boson contractions give the operator product expansionsThey form the level-one SU(2) Lie algebra, and there is an independent right-moving copy. The string vertex operators $J_L^a\bar{\partial }{X}^{\mu }$ and $\partial X^{\mu }{J}_R^a$ give the six massless vectors. ConsequentlyChanging $R$ gives the four charged vectors the mass found above, consistent with the radius deformation as a Higgs mechanism.

For completeness, the enhanced spectrum also has scalar fields. Using a compact rather than noncompact oscillator in the four singly excited charged families gives four scalars. At $(N_L,N_R)=(0,0)$, closed-string level matching requires $nw=0$; the mass equation then gives the four charges $(\pm 2,0),(0,\pm 2)$. These exceptional massless ground states of a bosonic circle give another four scalars. Together with the radius scalar they form nine scalars in the $(3,3)$ representation, in addition to the neutral dilaton. The ever-present neutral bosonic tachyon still has $M^2=-4/{\alpha }^{\prime }$. The enhancement establishes the additional massless vectors, not stability of the entire bosonic vacuum.

The RNS string adds worldsheet Majorana fermions ${\psi }^{\mu }$ to the coordinate fields $X^{\mu }$. The new fields are spinors on the worldsheet but vectors under the target Lorentz group; this distinction matters when interpreting their oscillator states. In a flat target, a convenient Lorentzian convention for the free gauge-fixed action iswhere ${\rho }^a$ are two-dimensional gamma matrices. The bosonic and fermionic kinetic terms are related by worldsheet supersymmetry. Before fixing local worldsheet supersymmetry, one introduces a worldsheet gravitino along with the worldsheet metric. After superconformal gauge fixing, the residual matter constraints are the Virasoro constraints and the constraints generated by the superconformal current $G\propto \psi \cdot \partial X$.

Each real worldsheet Majorana fermion contributes $1/2$ to the chiral central charge, while each coordinate contributes one. The reparameterization bc system contributes $-26$, and fixing local worldsheet supersymmetry also produces commuting superconformal ghosts of weights $(3/2,-1/2)$, with central charge $11$. ThusIn light-cone gauge in string theory there are eight physical transverse coordinate fields and eight transverse worldsheet Majorana fermions.

On the doubled spatial interval of an open string, the two allowed fermionic periodicities areThey arise from the relative signs of the left-right fermion gluing conditions at the two ends. Thus the Ramond sector has integer-mode oscillators $d_n$, including $d_0$, and the Neveu–Schwarz sector has half-integer oscillators $b_r$, $r\in \mathbb{Z}+1/2$, without zero modes. In either sector,A closed string has independent left- and right-moving periodicities, giving four sector choices rather than two.

The Ramond zero-mode Clifford algebra isHence its ground states carry a Spinor representation of the Lorentz group. The zero-mode supercurrent constraint reduces on a ground state to the Dirac equation $p_{\mu }{\Gamma }^{\mu }u=0$. In the Neveu–Schwarz sector the oscillator vacuum instead carries a scalar representation. Consequently the open string NS sector describes spacetime bosons, while the R sector describes spacetime fermions. Exciting a target-vector oscillator does not change this statistics assignment.

The string intercepts follow from the transverse zero-point energy. Per boson the integer-mode contribution is $\frac{1}{2}\zeta (-1)=-1/24$. Per fermion it is the negative of the corresponding half-sum. Since $\zeta (-1,1/2)=1/24$, the eight transverse pairs giveThus $a_{\mathrm{NS}}=1/2$, $a_{\mathrm{R}}=0$, and the open string mass formula is ${\alpha }^{\prime }{M}^2=N-a$. This distinguishes the lowest states explicitly:

The supersymmetric GSO projection keeps odd oscillator fermion number in the NS sector. Defining $F_{\mathrm{osc}}$ so that the NS vacuum has $F_{\mathrm{osc}}=0$, its projector isIt removes the scalar tachyon and keeps the massless vector. In the R sector, fermion parity contains the zero-mode chirality operator, so one may writeHere $F_{\mathrm{osc}}$ counts nonzero-mode fermionic excitations. On the ground states, this keeps one Majorana-Weyl spinor chirality: 16 real covariant components, or eight on-shell polarizations after the Dirac equation. The projected massless open-string states are an eight-polarization gauge vector and an eight-polarization gaugino. The compatible projected tower exhibits spacetime supersymmetry; unprojected worldsheet supersymmetry by itself did not eliminate the tachyon or equalize these massless counts.

For a closed string, both chiral zero-mode constraints must hold:This shifted closed-string level matching is particularly important in the mixed sectors. The lowest states and their spacetime nature are as follows.

For type IIA superstring theory, the two R sector ground-state chiralities are opposite. The Ramond–Ramond potentials can be represented by $C_1,C_3$, with transverse counts $8+56=64$. The two mixed-sector gravitinos have opposite chiralities, and the ten-dimensional theory is nonchiral. For type IIB superstring theory, the two R sector chiralities agree; the Ramond–Ramond potentials are $C_0,C_2,C_4$, with self-dual five-form field strength, giving $1+28+35=64$. Its two gravitinos have the same chirality. Thus both projected type II theories have 128 massless bosonic and 128 massless fermionic polarizations and no tachyon, while their chiralities and form spectra differ.

If unoriented open strings are included in the complete theory, the supersymmetric example is type I string theory. Worldsheet orientation reversal projects the closed type IIB superstring theory spectrum and identifies the two mixed sectors; the surviving closed fields include the graviton, dilaton, a Ramond–Ramond potential $C_2$, and one gravitino and dilatino. The open states form vector and gaugino multiplets with Chan-Paton factors; tadpole and anomaly cancellation select $SO(32)$. This is an additional orientation projection, not part of the GSO projection itself. The heterotic string instead pairs one supersymmetric chiral side with a bosonic chiral side carrying an internal gauge sector; it does not have a Ramond choice on both sides.

Take $0\le \sigma <2\pi$. Variation of the conformal-gauge action, with periodic variations, gives $({\partial }_{\tau }^2-{\partial }_{\sigma }^2)X^{\mu }=0$. Hence $X^{\mu }=X_R^{\mu }(\tau -\sigma )+X_L^{\mu }(\tau +\sigma )$. In an uncompactified direction, periodicity allows equal linear slopes and independent periodic parts. Expanding the periodic parts in a Fourier series yields the closed-string mode expansionHere ${\alpha }_m$ labels the right-moving sector and ${\tilde{\alpha }}_m$ the left-moving sector; exchanging the names is harmless if done consistently. Integration of the canonical momentum density $(2\pi {\alpha }^{\prime }{)}^{-1}{\dot{X}}^{\mu }$ over the string gives $p^{\mu }$, fixing the zero-mode coefficient. Reality gives ${\alpha }_m^{†}={\alpha }_{-m}$ and the analogous relation for tildes, with ${\alpha }_0={\tilde{\alpha }}_0=\sqrt{{\alpha }^{\prime }/2}p$.

The closed-string physical-state Virasoro conditions arewhere $L_0={\alpha }^{\prime }{p}^2/4+N$ and ${\tilde{L}}_0={\alpha }^{\prime }{p}^2/4+\tilde{N}$. Thus closed-string level matching gives $N=\tilde{N}$, and $M^2=-p^2=4(N-1)/{\alpha }^{\prime }$. The massless level is $N=\tilde{N}=1$. Its general state isThe string oscillator commutators give $[L_m,{\alpha }_n^{\mu }]=-n{\alpha }_{m+n}^{\mu }$: the two terms obtained by commuting through the quadratic generator coincide and cancel its factor $1/2$. Therefore $L_1$ replaces ${\alpha }_{-1}^{\mu }$ by ${\alpha }_0^{\mu }$, while $L_{m\ge 2}$ produces positive string oscillators that annihilate this level. The same holds independently for tildes. The complete polarization conditions areNull string states identify ${\zeta }_{\mu \nu }$ with ${\zeta }_{\mu \nu }+p_{\mu }{\lambda }_{\nu }+{\tilde{\lambda }}_{\mu }{p}_{\nu }$, with transverse $\lambda ,\tilde{\lambda }$. At nonzero momentum this leaves $24\cdot 24=576$ physical polarizations. Their symmetric traceless, antisymmetric and transverse-trace parts are the graviton, Kalb–Ramond field and dilaton.

The original PDF's condition ${\alpha }^{\prime }{p}^2=4$ is inconsistent with calling the displayed excited states massless. It is the on-shell condition of the string oscillator vacuum as a physical tachyon. For the level-one state, that value would give $L_0={\tilde{L}}_0=2$, violating the physical conditions. The ket $|0;p⟩$ used as the basis for a massless excitation must instead carry the excitation's momentum $p^2=0$; by itself it is then not an on-shell physical vacuum state.

For the compact coordinate, $X^1(\tau ,\sigma +2\pi )=X^1(\tau ,\sigma )+2\pi wR$, $w\in \mathbb{Z}$. Its momentum is $n/R$, $n\in \mathbb{Z}$, since the centre-of-mass wavefunction is single-valued on the circle. Thus its zero modes becomewithThese are momentum and winding modes. Let $k$ be the 25-dimensional noncompact momentum and $M^2=-k^2$. The two zero-mode constraints now giveTheir difference and average give, respectively,This also displays T-duality $R\leftrightarrow {\alpha }^{\prime }/R$, $n\leftrightarrow w$, with one chiral momentum reversed.

At a generic radius, the massless solutions are only $n=w=0$, $N=\tilde{N}=1$. Their fields in 25 dimensions are a graviton (275 polarizations), antisymmetric two-form (253), dilaton (one), two abelian vectors from $G_{a1}$ and $B_{a1}$ (23 each), and the radius scalar (one). Hence the generic massless spectrum has $275+253+1+46+1=576$ polarizations, with gauge group $U(1{)}_L\times U(1{)}_R$. Here generic excludes isolated radii where exceptional massless ground states of a bosonic circle occur. Indeed $N=\tilde{N}=0$ forces $nw=0$, and becomes massless at $R=|n|\sqrt{{\alpha }^{\prime }}/2$ or $R=2\sqrt{{\alpha }^{\prime }}/|w|$. These additional thresholds matter in the bosonic theory.

At the self-dual circle $R=\sqrt{{\alpha }^{\prime }}$, masslessness and level matching readSince $N,\tilde{N}\ge 0$, their sum is at most two. For sum two, $n=w=0$ and level matching gives the ordinary $(1,1)$ family. For sum one the four possibilities areThey have one right-moving ${\alpha }_{-1}$ or one left-moving ${\tilde{\alpha }}_{-1}$ excitation, as their levels specify. Each family has 24 physical polarizations: a 25-dimensional vector with 23 polarizations and one compact-oscillator scalar. Thus these families add 96 states.

For sum zero, level matching forces $nw=0$, while the mass equation requires $n^2+w^2=4$. It gives four additional scalar states with charges $(\pm 2,0)$ and $(0,\pm 2)$, without any string oscillators. They must not be omitted. The massless spectrum at the bosonic self-dual circle therefore has 100 additional polarizations, 676 in total. The four new vector fields enhance the gauge group to $SU(2{)}_L\times SU(2{)}_R$. Including the original radius scalar, the nine non-dilaton scalars form its $(3,3)$ representation. Equivalently, the full count is $275+253+1+6\cdot 23+9=676$.

Use units $k_B=\hslash =c=1$. In critical light-cone gauge in string theory, each chiral sector of the closed bosonic string has 24 transverse string oscillator species. Their generating function isEach factor sums the independent occupation numbers of the string oscillators at level $m$. For $q=e^{-t}$ with $t\downarrow 0$, the modular transformation of the Dedekind eta function givesExtract the coefficient by a contour integral and apply the saddle-point approximation to the exponent $(N-1)t+4{\pi }^2/t$. The saddle is $t_{\ast }=2\pi /\sqrt{N-1}$, with exponent $4\pi \sqrt{N-1}$ and second derivative $(N-1{)}^{3/2}/\pi$. The factor $t^{12}$ gives $(N-1{)}^{-6}$, and the Gaussian width gives $(N-1{)}^{-3/4}/\sqrt{2}$. HenceFor a noncompact closed string, closed-string level matching imposes equal left and right levels, but does not correlate their transverse polarizations. Therefore the large-level degeneracy of a closed bosonic string isThe mass relation is ${\alpha }^{\prime }{M}^2=4(N-1)$. Thus $8\pi \sqrt{N-1}={\beta }_{H}M$ withDistinguish degeneracy per discrete level from a smoothed density per unit rest mass. Since $dN/dM={\alpha }^{\prime }M/2$, the latter isThe exponential growth comes from the many ways of distributing string oscillator energy along a string; it is much faster than a fixed finite collection of particle species. The corresponding high-energy entropy is $S(E)={\beta }_{H}E+O(\log E)$, so its microcanonical temperature tends to $T_H$. Additional energy can go into very long strings rather than raising the temperature substantially.

The Boltzmann suppression competes with this exponential density. For the string oscillator rest-mass sum alone the large-$M$ tail is $\int dM{M}^{-26}{e}^{-(\beta -{\beta }_H)M}$. It converges for $\beta >{\beta }_H$ and fails for $\beta <{\beta }_H$. To include translational states in 25 noncompact spatial dimensions, take the one-string partition function per unit spatial volume. At large $M$,This follows by expanding the energy as $M+p^2/(2M)+\cdots$ and doing a Gaussian momentum integral. Its high-mass contribution is consequently proportional toThis integral is finite at $\beta ={\beta }_H$, although nonanalytic there: its first twelve derivatives in $\beta$ are finite, while the thirteenth diverges. The endpoint expansion contains a term proportional to $(\beta -{\beta }_H{)}^{25/2}$. Thus one must not assert that the canonical partition function necessarily diverges exactly at $T_H$; the polynomial prefactor and the spatial spectrum matter. This is the finite canonical partition function at the bosonic Hagedorn threshold distinction. The multi-string ideal gas has the same leading Hagedorn threshold, since the higher occupation-cycle terms probe $\ell \beta$, $\ell \ge 2$, and are nonsingular at ${\beta }_H$.

There is also a spacetime diagnosis. Euclidean temperature makes time a circle of radius $R_{\beta }=\beta /(2\pi )$. A string oscillator ground state winding once around that circle, with zero momentum number, hasIt becomes massless at ${\beta }_H$ and tachyonic for $T>T_H$. This thermal winding instability of a closed bosonic string signals failure of the perturbative low-temperature gas and motivates a transition to a different high-energy phase, often described in terms of long-string or winding-mode condensation. Free state counting fixes the threshold and its singular behavior, not the interacting phase's endpoint or the order of a transition.

Finally, the ordinary bosonic tachyon is already an infrared instability at zero temperature. The above canonical discussion is a formal analysis of the high-level spectrum with that separate infrared problem regulated. It does not establish a stable thermal vacuum of the unmodified bosonic theory; an actual equilibrium transition requires an appropriate stable setting and control of interactions.

The string sigma model in a spacetime metric is the Polyakov actionTake conformal gauge $h_{ab}=\mathrm{diag}(-1,1)$, and choose spatial period $2\pi$ for the closed string. Substitution of the plane-wave metric givesThe stipulated light-cone gauge in string theory $X^{+}=\tau$ sets ${\dot{X}}^{+}=1$ and $X^{\prime +}=0$. Its longitudinal term $-T\int {\dot{X}}^{-}$ is a total time derivative. The transverse string sigma model in an isotropic plane wave is thereforeThe resulting Euler-Lagrange equations are the massive wave equationsThese conventions fix the mass term directly from $X^{+}=\tau$. More generally $X^{+}=\kappa \tau$ would replace $m$ by $|m\kappa |$, with $p^{+}=2\pi T\kappa$ for this spatial period.

Use the real orthonormal Fourier basisFor the plane-wave string mode frequencies, let ${\omega }_0=|m|$ and ${\omega }_{n,c}={\omega }_{n,s}=\sqrt{n^2+m^2}$. The complete real mode expansion for nonzero frequencies isEvery term obeys the wave equation and is periodic. Orthonormality and completeness let its coefficients match arbitrary appropriate initial position and velocity data. For $m=0$, replace the zero-mode bracket by $q_0^I(0)+\tau p_0^I/T$; the zero mode becomes a free particle rather than an oscillator.

The canonical momentum density is ${\Pi }_I=T{\dot{X}}^I$. The transverse Hamiltonian isWrite $X^I={\sum }_a{q}_a^I{f}_a$ and $p_a^I=T{\dot{q}}_a^I$. Since $\int f_a{f}_b={\delta }_{ab}$ and $\int f_a^{\prime }{f}_b^{\prime }=n_a^2{\delta }_{ab}$, substitution gives the plane-wave string mode HamiltonianThis is a collection of independent harmonic oscillators. The last equality follows by substituting the sine-and-cosine time dependence; each oscillator energy is constant.